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Suppose the surface charge density over a sphere of radius R depends on a polar angle θ as σ = σ0 cos θ, where σ0 is a positive constant. Show that such a charge distribution can be represented as a result of a small relative shift of two uniformly charged balls of radius R whose charges are equal in magnitude and opposite in sign. Resorting to this representation, find the electric field strength vector inside the given sphere.

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We start from two charged spherical balls each of radius R with equal and opposite charge densities +ρ and -ρ. The centre of the balls are at  respectively so the equation of their surfaces are  

considering a to be small. The distance between the two surfaces in the radial direction at angle θ is | acos θ|  and does not depend on the azimuthal angle. It is seen from the diagram that the surface of the sphere has in effect a surface density σ = σ0 cos θ

Inside any uniformly charged spherical ball, the field is radial and has the magnitude given by Gauss’s theorm

In vector notation, using the fact the V must be measured from the centre of the ball, we get, for the present case

When vector K is the unit vector along the polar axis from which θ is measured.

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